{"id":"c5bfde4e-aeae-4040-8953-6f3975764de8","arxiv_id":"2412.02170","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper proposes a sign-parameterized formula that unifies the three common conventions for strong phase shifts in nonleptonic baryon decays and recomputes published phase shifts consistently.","lead":"This paper reviews how different experiments and theory groups define the strong phase shift in baryon decays, and proposes one unified formula that covers all common conventions. It tabulates the phase shifts for a set of published hyperon and charmed-baryon measurements under the new convention.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (10) as displayed lacks the square root in the denominator, making the central parameterization invalid as printed; the intended form is 2 arctan[(β sign)/(√(α²+β²)+α sign)], and the table only reproduces with that form.","rationale":"The key issue with the paper's central claim is that Eq. (10) as displayed is missing the square root in the denominator. The extracted text shows 'β× signp α2 +β2 +α× sign', which is unreadable without the square root; if taken literally, it is dimensionally inconsistent. The table's values confirm that the authors intended the half-angle formula with √(α²+β²). Because the paper's contribution is precisely this unified parameterization, the misprint is a load-bearing presentation error that prevents readers from applying the formula. The reader's weakest_assumption about exhaustiveness of conventions is not the most problematic; the paper explicitly scopes Eq. (10) to the three conventions discussed, and within those it is algebraically correct. I propose a direct numerical check to settle whether the printed formula matches the intended one. The reader's CONDITIONAL verdict is appropriate, and I do not change it.","tokens_in":6154,"tokens_out":26501,"duration_ms":258021,"concrete_test":"Recompute the E756 row (α=-0.458, β=-0.03, sign=+1) with the formula as printed in the manuscript text (no square root) and with the intended formula including √(α²+β²) in the denominator. Compare both outputs to the table's -3.08 rad; if the printed version does not match, the displayed equation is missing the square root and must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The full-text extraction of Eq. (10) reads 'δP−δS = 2 arctan β× signp α2 +β2 +α× sign', which omits the square root and the fraction bar. Taken literally, the denominator is α²+β²+α sign, and the formula does not compute the phase shift. For the E756 row (α=-0.458, β=-0.03, sign=+1), the literal expression gives about 0.24 rad, whereas the table lists -3.08±0.09 rad; the intended expression with √(α²+β²) gives -3.08 rad. Since Eq. (10) is the proposed unified parameterization, a reader using the displayed formula cannot reproduce the paper's own table or apply it to future studies. The same issue affects Eq. (4). This is a presentation defect rather than a conceptual flaw, but it is load-bearing because the central claim is the formula itself. The paper also does not prove that its three enumerated conventions exhaust the literature, but that is secondary.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reviews the conventions used in the literature for the strong phase shift δP−δS in two-body nonleptonic baryon decays, proposes Eq. (10) as a unified parameterization that accommodates three common amplitude parameterizations, and presents a table of phase shifts computed from published α, β, and ϕ values. The paper also discusses the impact of sign ambiguities on global averages and on CP-violation studies in baryon decays.","tokens_in":6362,"tokens_out":6102,"duration_ms":62072,"significance":"If the proposed formula and table are correct, the paper provides a compact and useful service to the baryon-CP community: it collects experimental results from different experiments, exposes the sign ambiguity, and offers a common language for comparing phase shifts. The underlying identity is standard, but the compilation and the explicit attention to the sign convention are useful. However, the central equation is not correctly displayed in the manuscript as provided, and the definition of the sign parameter is incomplete; both issues must be fixed before the paper can serve as a reliable reference.","major_comments":[{"comment":"The displayed form of Eq. (10), and likewise Eq. (4), is incomplete: it reads as 'δP−δS = 2 arctan β× signp α2 +β2 +α× sign', with no square root and no fraction bar. Taken literally, this expression is not the claimed phase-shift formula and cannot reproduce the values in Table 1. For example, in the E756 row (α=−0.458, β=−0.03, sign=+1) the literal expression gives about 0.24 rad, whereas the table lists −3.08±0.09 rad; the intended identity with √(α²+β²) in the denominator gives −3.08 rad. Because Eq. (10) is the central result, please correct the typesetting of Eqs. (4) and (10) and re-verify the table entries with the intended formula.","section":"Eqs. (4) and (10)"},{"comment":"The constant 'sign' is introduced without an explicit mapping to the three conventions. For convention 2, α2 and β2 in Eq. (8) are proportional to eS eP, and for convention 3, α3 and β3 in Eq. (9) are proportional to sin(2ζ); in convention 1 the coefficient is positive. The paper should state that 'sign' equals the sign of the common coefficient, namely sign(2|S||P|/(|S|²+|P|²))=+1, sign(2eS eP/(|eS|²+|eP|²)), or sign(sin 2ζ), respectively. Without this identification, a reader cannot determine which value of sign to use for a given published analysis, and the table's 'unknown' entries are ambiguous between a convention-dependent and experimental source.","section":"Definition of 'sign' in Eq. (10)"}],"minor_comments":[{"comment":"The text attributes Eq. (4) to Ref. [10] as 'the new formula', but the formula as shown is garbled; the same typographical correction needed for Eq. (10) applies here.","section":"Eq. (4)"},{"comment":"The column heading 'Value of sign' should be clarified to indicate whether the sign is inferred from the amplitude convention or explicitly reported by the experiment; for rows with no entry the paper should state explicitly that the sign is unknown.","section":"Table 1"},{"comment":"A one-line derivation of Eq. (10) using the identity tan(Δ/2)=sinΔ/(1+cosΔ) would help readers see why the formula is exact and why the sign choice changes δP−δS by π while leaving tan(δP−δS) invariant.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is more of a community-service compilation than a new physics result; its scientific content is sound in principle. My recommendation is driven by the two load-bearing presentation issues: the corrupted typesetting of the central formula and the undefined sign parameter. Both are readily fixable. If the authors also modestly acknowledge that their convention list is illustrative rather than exhaustive, the paper would be a useful reference for the baryon CP community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this is a short conventions/compilation note, not a physics advance. The proposed unified formula is a one-line modification of Eq. (4) from Ref. [10], and the numerical table is a recompilation of published measurements. That is fine—it is a genuinely useful service—but set expectations accordingly.\n\nWhat it does well: it collects the three common parameterizations of S and P amplitudes, shows how they differ by a sign, and provides a single formula with a 'sign' parameter. The table is handy: for each decay it lists α, β, ϕ, the original phase shift, and the value from the unified formula for both signs. It also correctly flags that E756's published δP−δS = 0.06±0.09 is inconsistent with its α<0 under convention (2); that is a real observation worth having. The π-shift property of the opposite sign choice is clearly explained, as is the fact that tan(δP−δS) is unaffected.\n\nThe soft spot is load-bearing and simple: the version of Eq. (10) I have in front of me reads 'δP−δS = 2 arctan β× signp α2 +β2 +α× sign' — that is, the square root in the denominator is missing (and the fraction bar is gone). Taken literally, the formula is wrong, and a reader cannot reproduce any row of the table. The table only works with the intended form '2 arctan[β·sign/(√(α²+β²)+α·sign)]'. The intended formula is standard and correct; the printed one is not. This must be fixed in the manuscript, and a referee should verify that the final typeset equation matches. The paper also asserts, without proof, that the three conventions in Eqs. (2),(6),(7) exhaust the literature; that is a minor gap, not a fatal one, but a sentence acknowledging possible other conventions would help.\n\nOverall: the paper is a useful reference for anyone combining baryon-decay measurements or planning CP-violation scans at BESIII/LHCb. It deserves a serious referee and publication after the equation typo is fixed. I would not describe it as a major contribution to the theory of CP violation; it is a tool, and a mostly good one.","headline":"Useful convention-compilation note whose central equation appears malformed as printed; fix the typo and it earns publication.","tokens_in":6883,"tokens_out":3162,"would_cite":false,"duration_ms":29886,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a single sign-parameterized formula, Eq. (10), that reproduces the strong phase shift $\\delta_P-\\delta_S$ under all three conventions used in the literature.","keywords":["strong phase shift","nonleptonic baryon decays","CP violation","hyperon decays","charmed baryon decays","polarization parameters","arctan ambiguity","decay asymmetry parameters"],"falsifier":"Find a published baryon-decay analysis whose S- and P-wave amplitudes are defined with a phase convention that cannot be reduced to Eqs. (2), (6), or (7), or show that with a known sign the phase shift from Eq. (10) disagrees with the paper's reported value by anything other than an integer multiple of $\\pi$; either result would disprove the claimed unification.","tokens_in":5965,"feed_emoji":"⚛️","tokens_out":11128,"duration_ms":95816,"temperature":0.7,"pith_summary":"This paper tries to establish that the strong phase shift $\\delta_P-\\delta_S$ in two-body nonleptonic baryon decays, a quantity that directly controls the size of CP-violating asymmetries, can be evaluated by a single unified formula regardless of which of three parameterization conventions a measurement uses. The three conventions are complex amplitudes with magnitudes $|S|$ and $|P|$, real amplitudes $e_S$ and $e_P$ that may be negative, and amplitudes written in terms of a mixing angle $\\zeta$. The paper proposes Eq. (10), a modified arctangent formula containing a sign constant, and shows that it reproduces all three conventions. Applied to published $\\Lambda$, $\\Xi$, and $\\Lambda_c$ data, the formula shows that opposite sign choices shift the reported phase shift by $\\pi$ while leaving $\\tan(\\delta_P-\\delta_S)$ unchanged. This gives a convention-independent way to combine phase-shift measurements from different experiments and to input them into future baryon CP-violation searches.","feed_headline":"One sign parameter unifies strong-phase conventions in baryon decays","feed_subtitle":"New formula maps three published conventions for δP−δS into one, ending sign ambiguities in CP-violation searches.","key_machinery":"The carrying object is the sign-parameterized arctangent formula, Eq. (10): $$\\delta_P-\\delta_S = 2\\arctan\\left(\\frac{\\$\\beta$\\,\\mathrm{sign}}{\\sqrt{\\$alpha^{2}$+\\$beta^{2}$}+\\$\\alpha$\\,\\mathrm{sign}}\\right).$$ The sign constant encodes the convention-specific sign of the product of the S- and P-wave amplitudes, or the sign of $\\sin(2\\zeta)$, which is often not stated in experimental papers. The formula works because $\\alpha$ is proportional to the cosine of the phase shift and $\\beta$ to its sine, and the denominator $\\sqrt{\\alpha^2+\\beta^2}+\\alpha\\,\\mathrm{sign}$ selects the correct branch of the arctangent, returning an angle in $[-\\pi,\\pi]$ rather than only in the principal branch $[-\\pi/2,\\pi/2]$. One formula thus replaces the case-by-case branch corrections previously applied convention by convention.","core_discovery":"The central claim is that the strong phase shift $\\delta_P-\\delta_S$ in any two-body nonleptonic baryon decay can be extracted from the measured $\\alpha$ and $\\beta$ by $$\\delta_P-\\delta_S = 2\\arctan\\left(\\frac{\\$\\beta$\\,\\mathrm{sign}}{\\sqrt{\\$alpha^{2}$+\\$beta^{2}$}+\\$\\alpha$\\,\\mathrm{sign}}\\right),$$ where 'sign' is a constant equal to either $+1$ or $-1$. For the first convention, $\\mathrm{sign}=+1$ reduces the formula to the expression recently proposed in Ref. [10]; for the second convention, sign is the sign of the product $e_S e_P$ of the real amplitudes; for the third, it is the sign of $\\sin(2\\zeta)$. The paper shows numerically, across the experiments collected in its table, that switching sign changes $\\delta_P-\\delta_S$ by $\\pi$ but leaves $\\tan(\\delta_P-\\delta_S)$ exactly unchanged, so the tangent is convention-independent while the angle itself carries a $\\pi$ ambiguity. It further points out that in one published $\\Xi^-\\to\\Lambda\\pi^-$ analysis the reported $(\\alpha,\\delta_P-\\delta_S)$ pair is internally inconsistent once this branch structure is respected.","pith_inferences":["If the same sign-parameterized formula is applied to future measurements in $\\Lambda_b$ and $\\Xi_b$ decays, the convention ambiguity should appear in the same two-branch form, because the standard S- and P-wave partial-wave decomposition has the same structure for any $1/2^+\\to 1/2^+ + 0^-$ decay.","The sign that resolves each measurement's branch can be compared with the sign of the product of S- and P-wave amplitudes predicted by a theoretical amplitude scheme, giving a consistency test that is independent of absolute phase conventions.","A practical rule suggested by the paper, but not stated by it, is that data tables should report $\\alpha$, $\\beta$, and the sign rather than a phase-shift value alone, since the latter is not convention-free."],"forward_implications":["The same measured pair $(\\alpha,\\beta)$ yields two possible phase shifts differing by $\\pi$, and $\\tan(\\delta_P-\\delta_S)$ is identical for both, so any global average that uses only the tangent is convention-independent.","Published $\\Lambda$, $\\Xi$, and $\\Lambda_c^+$ data can all be re-expressed in Eq. (10), with the sign usually recoverable from the original papers, as the table demonstrates.","The E756 result on $\\Xi^-\\to\\Lambda\\pi^-$, which reported $\\delta_P-\\delta_S=0.06\\pm0.09$ with $\\alpha<0$ under the first convention, is internally inconsistent; applying Eq. (10) gives $-3.08\\pm0.09$ rad for the same sign choice.","BESIII's two solutions for $\\Lambda_c^+\\to\\Xi^0K^+$, $-1.55\\pm0.25$ rad and $1.59\\pm0.25$ rad, correspond exactly to the two sign choices, so the measured large phase shift persists under the unified form.","Future experimental papers on baryon weak decays should quote the sign or explicitly state the amplitude convention, so that phase-shift values can be combined without bookkeeping errors."],"supporting_citations":[{"why":"Defines the partial-wave parameterization of α, β, and γ in terms of S- and P-wave amplitudes, the basis of the whole analysis.","marker":"[8]"},{"why":"Gives the standard definitions of the polarization parameters and the relation β = √(1−α²) sinϕ used in the helicity framework.","marker":"[9]"},{"why":"Introduced the earlier arctan formula for δP−δS (Eq. (4)) that the proposed Eq. (10) modifies by adding the sign constant.","marker":"[10]"},{"why":"Supplies the second parameterization with real eS and eP amplitudes and the CP-asymmetry formula A_CP = −tan(δP−δS) tan(ξP−ξS).","marker":"[11]"},{"why":"Reports the first measurement of a large phase shift in Λc+→Ξ0K+, the empirical anchor for the two sign-related solutions the paper explains.","marker":"[13]"},{"why":"Provides the third parameterization with amplitudes |A| sinζ and |A| cosζ, showing how sin(2ζ) controls the sign.","marker":"[16]"},{"why":"Supplies the E756 Ξ−→Λπ− data whose reported phase shift and α value are internally inconsistent, illustrating the arctan-branch issue the paper resolves.","marker":"[17]"},{"why":"Gives the LHCb measurements of nonzero phase shifts in Λc+→Λπ+ and Λc+→ΛK+, confirming large phase shifts in charmed baryon decays under the unified form.","marker":"[24]"}],"fun_headline_variants":["One sign constant unifies baryon decay phase conventions","New formula settles strong-phase ambiguity in baryon decays","Unified phase formula clears up baryon CP-violation inputs","Strong-phase conventions unified for baryon CP searches"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The unification assumes that every publication's convention is captured by one of the three parameterizations in Eqs. (2), (6), and (7), and that the relevant sign can in principle be identified; a fourth convention or a sign ambiguity that is not binary would break the formula.","fun_headline_variants_meta":{"raw":{"variants":["One sign constant unifies baryon decay phase conventions","New formula settles strong-phase ambiguity in baryon decays","Unified phase formula clears up baryon CP-violation inputs","Strong-phase conventions unified for baryon CP searches"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3274,"prompt_tokens":1064,"completion_tokens":2210,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":680,"completion_tokens_details":{"reasoning_tokens":2146}},"tokens_in":680,"tokens_out":2210,"duration_ms":17083,"temperature":1.0,"reasoning_tokens":2146,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:45:41.400464+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a published baryon-decay analysis whose S- and P-wave amplitudes are defined with a phase convention that cannot be reduced to Eqs. (2), (6), or (7), or show that with a known sign the phase shift from Eq. (10) disagrees with the paper's reported value by anything other than an integer multiple of $\\pi$; either result would disprove the claimed unification.","supporting_citations":[{"cited_title":"General partial wave analysis of the decay of a hyperon of spin 1/2, Phys Rev 1957;108:1645-1647","cited_arxiv_id":null,"evidence_quote":"Defines the partial-wave parameterization of α, β, and γ in terms of S- and P-wave amplitudes, the basis of the whole analysis."},{"cited_title":"Review of particle physics, Phys Rev D 2024;110:030001","cited_arxiv_id":null,"evidence_quote":"Gives the standard definitions of the polarization parameters and the relation β = √(1−α²) sinϕ used in the helicity framework."},{"cited_title":"Analysis of hadronic weak decays of charmed baryons in the topological diagrammatic approach, Phys Rev D 2024;109:114027","cited_arxiv_id":null,"evidence_quote":"Introduced the earlier arctan formula for δP−δS (Eq. (4)) that the proposed Eq. (10) modifies by adding the sign constant."},{"cited_title":"Hyperon decays and CP nonconserva- tion, Phys Rev D 1986;34:833","cited_arxiv_id":null,"evidence_quote":"Supplies the second parameterization with real eS and eP amplitudes and the CP-asymmetry formula A_CP = −tan(δP−δS) tan(ξP−ξS)."},{"cited_title":"First measurement of the decay asymmetry in the pure W-boson-exchange decay Λ+ c → Ξ0K+, Phys Rev Lett 2024;132:031801","cited_arxiv_id":null,"evidence_quote":"Reports the first measurement of a large phase shift in Λc+→Ξ0K+, the empirical anchor for the two sign-related solutions the paper explains."},{"cited_title":"Study of CP violation in hy- peron decays at super-charm-tau factories with a polarized electron beam, Phys Rev D 2022;105:116022","cited_arxiv_id":null,"evidence_quote":"Provides the third parameterization with amplitudes |A| sinζ and |A| cosζ, showing how sin(2ζ) controls the sign."},{"cited_title":"Measurement of decay parameters for Ξ−→ Λπ− decay, Phys Rev Lett 2003;91:031601","cited_arxiv_id":null,"evidence_quote":"Supplies the E756 Ξ−→Λπ− data whose reported phase shift and α value are internally inconsistent, illustrating the arctan-branch issue the paper resolves."},{"cited_title":"Measurement of Λ0 b, Λ+ c , and Λ decay parameters using Λ0 b→ Λ+ c h− decays, Phys Rev Lett 2024;133:261804","cited_arxiv_id":null,"evidence_quote":"Gives the LHCb measurements of nonzero phase shifts in Λc+→Λπ+ and Λc+→ΛK+, confirming large phase shifts in charmed baryon decays under the unified form."}],"review_version":1}